Final answer:
To calculate the sample size needed with a 0.02 margin of error at 90% confidence without an estimated proportion, we use the standard formula and determine that at least 3386 students should be sampled. None of the provided multiple-choice answers match this calculation, so the correct answer is D. cannot be determined.
Step-by-step explanation:
To determine the sample size needed for a survey with a specified margin of error and confidence level, we can use the formula for a confidence interval for a population proportion. The general formula for calculating the required sample size (n) is given by:
n = (z^2 * p * (1-p)) / E^2
Where:
- z is the z-score related to the confidence level
- p is the estimated proportion of college students working (if unknown, p is typically taken to be 0.5 for a conservative estimate)
- E is the margin of error
In this case, we want to estimate the proportion within a margin of error of 0.02 with 90% confidence. For 90% confidence, the z-score is approximately 1.645. If we do not have a preliminary estimate for the proportion (p), we use 0.5 for the most conservative sample size.
Plugging the values into the formula:
n = (1.645^2 * 0.5 * (1-0.5)) / 0.02^2
n = (2.708025 * 0.5 * 0.5) / 0.0004
n = 1.3540125 / 0.0004
n = 3385.03
Since we cannot survey a fraction of a person, we round up to the nearest whole number, so we need to sample 3386 college students.
Looking at the multiple-choice answers provided, none of them match the calculated sample size of 3386. Therefore, the correct multiple-choice answer to the how many college students should be sampled to meet the criteria is:
D. cannot be determined